Topics in Tame Geometry and Analysis

Mathematical analysis is concerned with the change behavior of functions. It grew from the attempt to describe and predict the physical world and is a fundamental tool of the natural sciences. Of central interest is the notion of derivative, i.e., the rate of change, and furthermore questions of differentiability, regularity, dependence on initial conditions, how much local properties determine global properties, etc. We will investigate fundamental problems at the interface between analysis and semialgebraic geometry, the study of sets of real solutions of polynomial equations and inequalities. Polynomials are one of the most basic objects in mathematics. Being rich enough to describe a large family of sets but also rigid enough so that semialgebraic sets and functions are tame in many respects, makes semialgebraic geometry very useful in a wide range of applications. A basic question in analysis, going back to Whitney in 1934 and solved 70 years later by Fefferman, is the extension problem: how can one tell if a function on a set extends to a differentiable function on the ambient space? We will study to what extent semialgebraic geometry is preserved: given a semialgebraic function having a differentiable extension, does it have a semialgebraic differentiable extension? Can characteristic semialgebraic parameters such as the number of polynomial equations and inequalities and their degrees be controlled? At the core of many analytical problems is the question of the regularity of the solutions of algebraic equations depending on parameters (e.g. the eigenvalues of linear operators). In recent years, we determined the optimal regularity and showed that the solution map is bounded w.r.t. to the natural structures on the spaces of coefficients and solutions. We will now study the continuity of the solution map: does a small variation of the coefficients of the equation induce a small variation of the solutions (relative to the natural structures as for boundedness)? In general, differentiable functions can have wild behavior. But differentiable functions with controlled growth of their derivatives often exhibit polynomial-like behavior such as effective bounds on the size of the zero set or powerful inequalities for the maximum value of the function on its domain in terms of its maximum value on a subset. So far this has been established for functions on convex domains. In the project, we will deepen the understanding of this striking similarity to polynomials and study these results on domains with more complicated geometry. Powerful theories of global analysis are based on the fact that often regularity of maps can be detected by composition with smooth curves. This generalizes to functions on certain tame closed domains with singular boundaries. We will study its delicate dependence on the geometric and analytic properties of the domain and the functional-analytic aspects of the respective function spaces.